Stabilization in a two-species chemotaxis system with a logistic source

Abstract

We study a system of three partial differential equations modelling the spatio-temporal behaviour of two competitive populations of biological species both of which are attracted chemotactically by the same signal substance. More precisely, we consider the initial-boundary value problem for under homogeneous Neumann boundary conditions in a bounded domain , n ⩾ 1, with smooth boundary.When 0 ⩽ a1 < 1 and 0 ⩽ a2 < 1, this system possesses a uniquely determined spatially homogeneous positive equilibrium (u⋆, v⋆). We show that given any such a1 and a2 and any positive diffusivities d1 and d2 and cross-diffusivities χ1 and χ2, this steady state is globally asymptotically stable within a certain nonempty range of the logistic growth coefficients μ1 and μ2.